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The number of representations of a natural number n as the sum of four squares of integers is denoted by r 4 (n). Jacobi's four-square theorem states that this is eight times the sum of the divisors of n if n is odd and 24 times the sum of the odd divisors of n if n is even (see divisor function), i.e.
In particular, for a prime number p we have the explicit formula r 4 (p) = 8(p + 1). [2] Some values of r 4 (n) occur infinitely often as r 4 (n) = r 4 (2 m n) whenever n is even. The values of r 4 (n) can be arbitrarily large: indeed, r 4 (n) is infinitely often larger than . [2]
The squared Euclidean distance between two points, equal to the sum of squares of the differences between their coordinates; Heron's formula for the area of a triangle can be re-written as using the sums of squares of a triangle's sides (and the sums of the squares of squares) The British flag theorem for rectangles equates two sums of two ...
Every non-negative real number is a square, so p(R) = 1. For a finite field of odd characteristic, not every element is a square, but all are the sum of two squares, [1] so p = 2. By Lagrange's four-square theorem, every positive rational number is a sum of four squares, and not all are sums of three squares, so p(Q) = 4.
Lagrange's four-square theorem states that every positive integer can be written as the sum of at most four squares. [ 5 ] [ 6 ] Four is one of four all-Harshad numbers . Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. 4 x = y 2 − z 2 {\displaystyle 4x=y^{2}-z^{2}} .
In mathematics and statistics, sums of powers occur in a number of contexts: . Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities.
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Division is also not, in general, associative, meaning that when dividing multiple times, the order of division can change the result. [7] For example, (24 / 6) / 2 = 2 , but 24 / (6 / 2) = 8 (where the use of parentheses indicates that the operations inside parentheses are performed before the operations outside parentheses).